I send an image to CS6, bring back into LR and wish to rename it to keep track of what I did. Go to library-> Rename image. What comes up in the custom renaming window is the name of the last image I worked on. This has no relation to image I am working on now. I wish it would bring up the name of the current image I am working on and allow to me change the name, instead of having to enter the entire name of the current image

Why not merely "save as" when finished with edits in CS6? Name it anything you like. I do this routinely. Sometimes entirely new name, sometimes suffix to orig that is meaningful re edits.

Added note for clarity - when you do as above, the file name of the round-tripped file appears as you "saved as" in CS6, in LR. No need to name change from within LR & less confusing IMO. Most often "save as" is used when saving as a new file type, but you may just leave as tiff and only change the file name.

If you select Edit, you'll get a floating menu that lets you choose which fields should be used in renaming. After you,ve created a new preset in this way, you can rename and save it by clicking on filename at the top of the menu.I'm not sure whether it will do exactly what you want, I rarely use th feature as I can't get it to do exactly what I want